Squirrelmail is a mail client, it does not store any mails. you can use multiple mail clients with your mail account or even exchange them at any time, this has no influence on your email.

The email itself is stored independantly form the client in /var/vmail. So all you have to do is to restore /var/vmail and the dbispconfig database from the old server to have all your mails and the complete mails etup back. Which mail client you use does not matter here.